Accurately modeling solid-state dewetting in materials with strong crystalline anisotropy is an open problem in materials science with importance for both the manufacturing and reliability of nano-scale devices. In this work, we propose and demonstrate a level-set method framework for simulating solid-state dewetting which is capable of modeling systems with strongly anisotropic surface energies and diffusivities. Surface energy anisotropy is handled through the use of the Cahn-Hoffman vector construction, and surface self-diffusivity anisotropy is handled through the use of a diffusivity tensor. We benchmark our method against isotropic phenomena with analytical descriptions and go on to demonstrate that our method is capable of reproducing a host of experimentally observed behaviors in strongly anisotropic single-crystal materials.